Mathematics methods and models

Research in this area is oriented towards the mathematical modelling, both from a theoretical and a numerical point of view, of a variety of phenomena arising in different areas of science, engineering, business, and industry. Typical applications include, for instance, quantum and classical fluid dynamics (nonlinear Schrödinger equations, advection-diffusion-reaction equations, nonlinear hyperbolic systems), complex multi-agent systems (kinetic equations, particle methods, multi-scale models), superconductivity and materials science (Ginzburg-Landau equations), evolution of interfaces in physics and biology (minimal surface equations, motion by mean curvature), image processing (total variation-based methods), optimal control (Hamilton-Jacobi equations), mathematical finance (stochastic differential equations, Lévy processes, stochastic control), statistical mechanics (interacting particle systems, stochastic systems with many degrees of freedom), classical and quantum mechanics. This area is characterised by a sophisticated integration of multidisciplinary skills in Analysis of nonlinear PDEs, Calculus of Variations, Optimal Control, Numerical methods for PDEs, Scientific Computing, Stochastic Analysis, Probability Theory, Mathematical Physics, and Differential Geometry. The research is carried out through several collaborations, partnerships, networks, and projects both at national and international levels. This area can be further divided into sub-areas, which are recognisably different but interact with one another. Analysis of PDEs and Calculus of Variations Keyphrases: nonlinear partial differential equations, geometric measure theory, optimal transport, variational models in superconductivity theory and materials science, control theory, mean field games. Numerical Analysis and Scientific Computing Keyphrases: numerical methods for hyperbolic systems, Finite Volume and Discontinuous Galerkin schemes, exponential integrators and matrix function approximation, numerical methods for nonlinear Schrodinger equations, numerical methods for kinetic equations, modelling of multi-agent systems. Probability and Mathematical Physics Keyphrases: complex systems, interacting particle systems, stochastic dynamics, mean field games, stochastic partial differential equations, mathematical finance, neural networks and applications, integrable systems, geometric methods in mechanics.